Differential invariants and invariant differential equations

Our objective in this workshop is to bring together a selected group of geometers, group theorists and analysts who are currently active in the field of differential invariants and invariant differential equations, as well as some researchers more directly involved in applications, so as to create a unique opportunity for the exchange of ideas on open problems which are central in the field.

On the geometric side, the timeliness of the proposed workshop is underscored by a newly established foundation for the method of moving frames, due to Fels and Olver ( see the references below ), which is a far-reaching and extraordinarily powerful generalization of the classical theory of Elie Cartan. This recent advance was inspired, on the one hand, by contemporary attempts to rigorize Cartan's methods, and, on the other, by the promise of significant applications in computer vision. It has made it possible to tackle important open problems such as the classification of differential invariants and invariant differential equations for a number of group actions of fundamental importance. The resulting theory and attendant computational algorithms go far beyond the geometrical contexts considered to date.

On the analytic side, we feel that in view of the spectacular recent advances in the analytic study of the mean curvature flow for submanifolds of Euclidean space and of smoothing by the flows associated to some non-linear invariant diffusion equations, the time is ripe for a small-scale meeting in which analysts and geometers will be brought together. Indeed, the connections between integrable soliton equations, Poisson geometry, and geometric motions of curves and surfaces can be traced back to the remarkable Hasimoto transformation between the equation of motion of a vortex filament and the nonlinear Schr\"odinger equation. Hierarchies containing both curve shortening flows, vortex filament equations, equations of elastic rods, and thermal grooving have now been established. The past decade has seen a steady increase in the range of theoretical developments, and practical applications in fluid mechanics, elasticity, geometry, computer vision, and soliton theory. Recent work has highlighted the role of differential invariants and invariant evolution equations in understanding and extending this connection to other geometries and other transformation groups. In particular, operators appearing in the invariant Euler-Lagrange complex constructed with the moving frame theory govern the bi-Hamiltonian structure, and hence integraility of these geometrical motions. However, a complete understanding of this connection remains unclear. The following is a list of references which are directly relevant to the objectives of the workshop: Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M., Axioms and fundamental equations of image processing, Arch. Rat. Mech., 123 (1993)

Alvarez, L., Lions, P.L., Morel, J.M., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992) 845--866

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Boutin, M., Numerically invariant signature curves, Int. J. Computer Vision, 40 (2000) 235--248

Ceyhan, \"O, Fokas, A.S., G\"urses, M., Deformations of surfaces associated with integrable Gauss--Mainardi--Codazzi equations, J. Math. Phys., 41 (2000) 2251--2270

Chern, S.S., Tenenblat, K., Pseudo--spherical surfaces and evolution equations, Stud. Appl. Math., 74 (1986) 55--83

Fokas, A.S., Symmetries and integrability, Stud. Appl. Math., 77 (1987) 253--299

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Fefferman, C., Graham, C.R., Conformal invariants, \'Elie Cartan et les Math\'ematiques d'aujourd'hui, Ast\'erisque, hors s\'erie, Soc. Math. France, Paris, 1985, pp. 95--116

Fels, M., Olver, P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math., 51 (1998) 161--213

Fels, M., Olver, P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math., 55 (1999) 127--208

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Gonz\'alez--L\'opez, A., Hern\'andez, R., Mar\'\i --Beffa, G., Invariant differential equations and the Adler--Gel'fand--Dikii bracket, J. Math. Phys., 38 (1997) 5720--5738

Griffiths, P.A., On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J., 41 (1974) 775--81

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Heredero, R.H., Olver, P.J., Classification of invariant wave equations, J. Math. Phys., 37 (1996) 6414--6438

Kamran, N., Milson, R., Olver, P.J. , Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, Adv. in Math., 156 (2000) 286--319

Kamran, N., Tenenblat, K., On differential equations describing pseudospherical surfaces, J. Diff. Eq., 115 (1995) 75--98

Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., Yezzi, A., Conformal curvature flows: from phase transitions to active vision, Arch. Rat. Mech. Anal., 134 (1996) 275--301

Hasimoto, H., A soliton on a vortex filament, J. Fluid Mech., 51 (1972) 477--485

Langer, J., Recursion in curve geometry, New York J. Math., 5 (1999) 25--51

Langer, J., Perline, R., Poisson geometry of the filament equation, J. Nonlin. Sci., 1 (1991) 71--93

Langer, J., Singer, D.A., Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev., 38 (1996) 605--618

Mar\'\i--Beffa, G., Olver, P.J., Differential invariants for parametrized projective surfaces, Commun. Anal. Geom., 7 (1999) 807--839f nonlinear partial differential equations to dynamical systems, Adv. in Math., 156 (2000) 286--319

Mikhailov, A.V., Shabat, A.B., Yamilov, R.I., The symmetry approach to classification of nonlinear equations. Complete lists of integrable systems, Russian Math. Surveys, 42{\rm :4} (1987) 1--63

Morel, J.--M., Solimini, S., Variational Methods in Image Segmentation, Progress in Nonlinear Diff. Eqs., vol. 14, Birkh\"auser, Boston, 1995

Mundy, J.L., Zisserman, A. (eds.), Geometric Invariance in Computer Vision, The MIT Press, Cambridge, Mass., 1992

Olver, P.J., Sapiro, G., Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math., 57 (1997) 176--194

Osher, S.J., Sethian, J.A., Front propagation with curvature dependent speed: Algorithms based on Hamilton--Jacobi formulations, J. Comp. Phys., 79 (1988) 12--49

Sanders, J.A., Wang, J.P., On the integrability of homogeneous scalar evolution equations, J. Diff. Eq., 147 (1998) 410--434

Sapiro, G., Tannenbaum, A., On invariant curve evolution and image analysis, Indiana Journal of Mathematics, 42 (1993) 985--1009

Sapiro, G., Tannenbaum, A., On affine plane curve evolution, J. Func. Anal., 119 (1994) 79--120

Tenenblat, K., Transformations of Manifolds and Applications to Differential Equations, Pitman Monographs and Surveys Pure Appl. Math., vol. 93, Addison Wesley, Longman, England, 1998

Tresse, A., Sur les invariants diff\'erentiels des groupes continus de transformations, Acta Math., 18 (1894) 1--88

Confirmed Participants

Abstracts (PDF)

Programme (PDF)

Videos

Final Report (PDF)